Stability analysis of time-varying perturbed descriptor systems. (English) Zbl 1539.37033
Summary: The manuscript deals with the analytical study of the “practical” stability of linear time dependent descriptor perturbed systems. The paper works out and proves the sufficient conditions (posed as LMIs) for practical stability of perturbed singular systems. The Gamidov’s type integral inequality is used as a tool in the proof of the sufficient conditions of practical stability for perturbed singular systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.
MSC:
| 37C75 | Stability theory for smooth dynamical systems |
| 34H15 | Stabilization of solutions to ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 37N35 | Dynamical systems in control |
Keywords:
linear time-varying descriptor systems; perturbed systems; standard canonical form; consistent initial conditions; Gamidov’s inequality; Lyapunov techniques; practical stabilitySoftware:
MORLABReferences:
| [1] | T. Berger, Bohl exponent for time-varying linear differential-algebraic equations, International Journal of Control, 10, 1433-1451, 2012 · Zbl 1253.93113 · doi:10.1080/00207179.2012.688872 |
| [2] | T. A. Berger Ilchmann, On the standard canonical form of time-varying linear DAEs, Quarterly of Applied Mathematics, 71, 69-87, 2013 · Zbl 1283.34007 · doi:10.1090/S0033-569X-2012-01285-1 |
| [3] | T. A. Berger Ilchmann, On stability of time-varying linear differential-algebraic equations, International Journal of Control, 86, 1060-1076, 2013 · Zbl 1278.93181 · doi:10.1080/00207179.2013.773087 |
| [4] | A. W. BenAbdallah Hdidi, Improving the performance of semiglobal output controllers for nonlinear systems, Kybernetika, 53, 296-330, 2017 · Zbl 1424.93096 · doi:10.14736/kyb-2017-2-0296 |
| [5] | A. W. BenAbdallah Hdidi, Uniting two local output controllers for linear system subject to input saturation: LMI approach, Journal of the Franklin Institute, 355, 6969-6991, 2018 · Zbl 1398.93132 · doi:10.1016/j.jfranklin.2018.08.001 |
| [6] | A. I. M. A. Benabdallah Ellouze Hammami, Practical stability of nonlinear time-varying cascade systems, Journal Dynamical Control Systems, 15, 45-62, 2009 · Zbl 1203.93160 · doi:10.1007/s10883-008-9057-5 |
| [7] | A. M. A. M. Ben Makhlouf Hammami Hammi, A new approach for stabilization of control affine systems via integral inequalities, IMA Journal of Mathematical Control and Information, 39, 837-860, 2022 · Zbl 1498.93554 · doi:10.1093/imamci/dnac007 |
| [8] | P. S. W. R. Benner Werner, Model reduction of descriptor systems with the MORLAB toolbox, IFAC-PapersOnLine, 51, 547-552, 2018 |
| [9] | S. L. Campbell, Singular Systems of Differential Equations, Res. Notes in Math., 40. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. · Zbl 0419.34007 |
| [10] | S. L. Campbell, Singular Systems of Differential Equations. II, Res. Notes in Math., 61. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. · Zbl 0482.34008 |
| [11] | T. F. M. A. Caraballo Ezzine Hammami, On the exponential stability of stochastic perturbed singular systems in mean square, Applied Mathematics and Optimization, 84, 2923-2945, 2021 · Zbl 1472.93152 · doi:10.1007/s00245-020-09734-8 |
| [12] | T. F. M. A. Caraballo Ezzine Hammami, New stability criteria for stochastic perturbed singular systems in mean square, Nonlinear Dynamics, 1, 241-256, 2021 · Zbl 1537.93533 · doi:10.1007/s11071-021-06620-y |
| [13] | T. F. M. A. Caraballo Ezzine Hammami, A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 116, 1-24, 2022 · Zbl 1500.93141 · doi:10.1007/s13398-022-01320-7 |
| [14] | M. Corless, Guaranteed rates of exponential convergence for uncertain systems, Journal of Optimization Theory and Applications, 64, 481-494, 1990 · Zbl 0682.93040 · doi:10.1007/BF00939420 |
| [15] | L. Dai, Singular Control Systems, Lect. Notes Control Inf. Sci., 118. Springer-Verlag, Berlin, 1989. · Zbl 0669.93034 |
| [16] | D. Lj. B. V. Debeljkovic Jovanović Drakulić, Singular system theory in chemical engineering theory: Stability in the sense of Lyapunov: A survey, Hemijska Industrija, 6, 260-272, 2001 |
| [17] | F. M. A. Ezzine Hammami, Growth conditions for the stability of a class of time-varying perturbed singular systems, Kybernetika, 58, 1-24, 2022 · Zbl 1524.34032 · doi:10.14736/kyb-2022-1-0001 |
| [18] | S. G. Gamidov, Certain integral inequalities for boundary value problems of differential equations, Differentsialnye Uravneniya, 5, 463-472, 1969 · Zbl 0167.08401 |
| [19] | S. F. C. M. H. Hafstein1 Kellett Li, Computing continuous and piecewise affine lyapunov functions for nonlinear systems, Journal of Computational Dynamics, 2, 227-246, 2015 · Zbl 1345.93119 · doi:10.3934/jcd.2015004 |
| [20] | V. Lakshmikantham, S. Leela and A. A. Martynyuk, Practical Stability of Nonlinear Systems, World Scientific, 1990. · Zbl 0753.34037 |
| [21] | H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992. · Zbl 0969.34001 |
| [22] | P. V. Kokotovic, A. Bensoussan and G. L. Blankenship, Singular Perturbations and Asymptotic Analysis in Control Systems, Lecture Notes in Control and Information Sciences, 1987. · Zbl 0605.00020 |
| [23] | D. H. D. Lj. Ownes Debeljković, Stability of linear descriptor systems: A geometric analysis, IMA Journal of Mathematical Control and Information, 2, 139-151, 1985 · Zbl 0637.93051 |
| [24] | Q. C. N. J. E. Pham Tabareau Slotine, A contraction theory approach to stochastic Incremental stability, IEEE Transcactions on Automatic Control, 54, 816-820, 2009 · Zbl 1367.60073 · doi:10.1109/TAC.2008.2009619 |
| [25] | E. D. Sontag, Input to state stability: Basic concepts and results, Lecture Notes Math, 12, 163-220, 2008 · Zbl 1175.93001 · doi:10.1007/978-3-540-77653-6_3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
